DOCTORAL SEMINAR 3
DOCTORAL SEMINAR 3 FNCE 7550
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This 3 page Class Notes was uploaded by Alvena Marks IV on Thursday October 29, 2015. The Class Notes belongs to FNCE 7550 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 25 views. For similar materials see /class/231810/fnce-7550-university-of-colorado-at-boulder in Finance at University of Colorado at Boulder.
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Date Created: 10/29/15
FNCE 7550 notes Garland B Durham Leeds School of Business University of Colorado October 22 2008 HW5 Consider the model 9 My 039Y eXPvt2 1r 1 H39Util O39VEQt With My 0 O39y 008 H 96 UV 5 en N N0l for i 12 corr 611621 0 and v1 N N0a17 H2 We observe the data y y1iuyn 024 0i17 093 016 044 i071 006 The states V 11 i vn are unobserved The goal is to compute the loglikelihood My My V W We Will compute this integral using Monte Carlo integration With the importance sampler 4 ie Lltygt pltyvgtdv dc2ltvgt Where q is described belowi The importance sampler Will be based on the Laplace approximation to the log target density logpbn V Zlogpmlvi 10gpv1 Z logpvilvi71A i1 i2 First construct C BBV logpyv and Q 7828V2 logpy V Where 8 Ci i 10gPyilU1 10gPv1 10g10v2lv1l avl a on g logpynlvn logpvnlvn71 E Ci av 10gPyilvi 10g10vilvi71 10g10vi1lvil7 1 27 i 7n 1 and Q is tridiagonal With elements QM 7aiwci il in 82 7m logpvilvi1 z 2 i ni 913121 9pm Now we need to nd the mode of the target density quotI Initialize by setting quotI 0 and then iterate several times using Newton7s method ie v v mlci where Q and C are evaluated at the current value of quotI on each iteration The importance sampler is NW 9 1 where Q is the negative Hessian as described above evaluated at A To draw from this let R be the Cholesky decomposition of Q and 55 be a draw from the multivariate standard normal with dimension Then a draw from the sampler is obtained by solving Vs 13515 I Given a sequence of draws Vs s l i i i 5 obtained in this way we can compute 5 VM My pyyvdv using qltvltsgtgt lt5lt5gtMRL where is the pdf of the multivariate standard normal and lRl the determinant of R is just the product of its diagonal elements Note that pyv5 can be obtained directly from the model de nitioni H Using the model parameters and observed y as above what is I E0 What is 9 evaluated at y y 3 What is 5 evaluated at y y 4 Using 5 100 draws from the importance sampler what is your estimate for Ly Repeat this several times to get a feel for the simulation errori Also try it several times using 5 1000i
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